The goal of this research project is to develop the coalescent theory to the level so that power of the information of DNA polymorphism, the ultimate resolution of genetic variation, can be fully realized. The theory has wide applications ranging from studying mutation rate to the estimation of the time to the common ancestor of our human beings. Specific aims include: a. to develop the theory of coalescent for the frequencies of segregating types. We propose to derive analytically the mean,, variances and covariances of the frequencies of segregating types under models, from the classic which assumes no recombination and equal rates of mutations for all sites to more realistic model that allows both recombination and unequal rates of mutations at different sites of sequences. The theory will make the application of the theory of coalescent highly efficient. b. to develop highly efficient statistical methods for estimating theta. We propose a framework, which will utilize both the phylogenetic information and the frequencies of segregating types, to estimate the crucial parameter theta=4Nemu. c. to develop powerful statistical method for testing the hypotheses concerned about theta. We propose to develop statistical tests for the assumptions underlying the coalescent theory, including the assumption of neutrality of mutations, by utilizing the information contained in frequencies of segregating types. d. to develop a visual method for identifying the main causes of genetic variations at DNA sequences. We propose to use graphical comparison of the observed frequencies of segregating types with the expected frequencies of segregating types under different assumptions, including selection, population subdivision. e. to estimate the time to common ancestor and determine the pattern and speed of population growth. We propose a maximum likelihood method for estimating the time to the most recent common ancestor of the individuals in a sample. This likelihood method also allows one to analyze the pattern and speed of changes in the effective population size.